I know you have seen a lot of this problem, but here it is one more time: suppose I have an isosceles triangle where the length of two sides is 64, and the angle between these sides measures 36 degrees. What is the area of this triangle?

By now, students are pretty familiar with this problem. Now that we have used isosceles triangles as tools for analyzing polygons inscribed in circles, I want to come back to this one more time in a final effort to drive home the idea that abstract expressions are actually a tool that allow us to achieve greater precision. What I’m looking for are two different students to share their work – I’d love to find one who rounds the base and height to the nearest tenth (I mean, even better would be someone who rounds to the nearest whole number, but I think they all know better than that at this point…), and another who uses the formula they developed yesterday for the area. The former student would have a solution of (0.5)(39.6)(60.9) = 1205.82, and the latter would have 1203.78417: a small difference, but noticeable enough for the students to see (area rounded vs formula). The idea is not to say that rounding is “wrong,” just that it changes an answer. In the context of this project, if we are pursuing π to the maximum number of decimal places, then this would be a big difference.

The purpose of today's opener is to provide a specific example of the kind of precision students might address as they get to writing their papers for the Defining Pi Project (MP6). As we look at this example, I ask students to consider the idea that context matters when we talk about precision. I’ve mentioned this in earlier lessons – what I might say to students here is that if I loaned my little brother $1205.82 to buy his first car, and he paid me back $1203.78, would I really mind that? But that’s not what we’re doing here, so I continue: we’re engaging in the historic exercise of calculating π as precisely as possible, with a healthy (and lighthearted) view that sometimes a trivial exercise can be most engaging.

Resources (1)

Resources

With that, students now have an hour of workshop time to spend as needed to get the project done. This could take any shape:

For those who have kept up, it means writing the best possible papers for Part 5.

For others, it might mean going back and getting one more construction done (Part 2) or finishing the Gallery Walk (Part 1).

One thing that I like about this project is that it makes more sense as it moves along. If students haven’t kept up, they at least have the concept that we’re trying to approximate a value for π by inscribing polygons, using isosceles triangles, and moving from specific constructions to abstracted formulas. When they go back to complete earlier parts of the project, they are able to see a wide variety of connections between the material. Today, my job is to move around the room and encourage/watch for/engage in discussions about these connections and insights.

A Few Things to Look For

Error checking takes center stage – have fun with it! The π Paper asks students to use the Excel spreadsheet to check some of their answers for earlier parts of the project. The point is not about being right or wrong, becauseeveryone should have a few values on Parts 2 or 4 that don’t match the spreadsheet perfectly. The point is that we want to figure out why these errors happened: was it rounding? A misunderstanding? A lack of parentheses? Something else?

A juicy (but non-essential) detail is that of how precise Excel is set to be. I don’t want to give this away, but I’d love for a few kids to figure it out, and then to look into methods/technology for being even more precise than this.

If it happens naturally, a discussion that ties things back to similarity can be great. Look for students asking something along the lines of, "if the radius is smaller, the area of the polygon is smaller, right?" The idea is that, yes, of course the polygon is smaller, but so is the circle, so the ratio of one to the other will be the same for any radius that you or your group-mates choose. This is how we defined similarity at the start of the semester.

Some students will get to the point of asking, “What if I put a decimal number in the first column?” Encourage this! Let them play with it. A “decimal” can be in the form of an angle like 22.5 degrees (in the middle of our list), or one like 0.5, 0.1 or 0.001 degrees, which is what we’d have to do to get the most precise value for π. What central angle is necessary to calculate π to as many decimal places as Excel can take?

Self-Assessment

The Self Assessment for the project provides students with a space to provide evidence of how they have met seven Mathematical Practices and two content learning targets. For my teaching practice, I find this part most important because it gives me a window into what kids really understand both about the meaning of the practices and about what they have done on this project. I make sure to emphasize to each table – and occasionally to specific students – that this is an important part of the project. For better or for worse, the idea that this is the part of the project where their grades are determined increases student buy-in. Up until this part, we’ve been blissfully ignorant of grades, and able to engage in an intellectual pursuit in spite of that!

Resources

We’ve spent the last eight class periods working through this project, and here ends our classwork time. Students still have a few days to get the project done, and I’ll be available to them outside of class time, but next class we’ll be moving on to formalizing some knowledge about arc length and sector area.

To get a quick snapshot of what students understand so far, I give everyone an index card, and ask them to write their name and to answer this simple question:

What was this project about?

This is also the first question on the paper students must write for Part 5, so some students will have already thought through their answer to it. I want to know where they stand right now, and what kinds of additional scaffolds I might have to add to the next few classes while students are finishing up on the project outside of class.